3.1.2 Rates and Unit Rates

Calculating rates as ratios of different units and finding unit rates to compare quantities per single unit.

定义

A rate is a ratio that compares two quantities measured in different units. It expresses how much of one quantity corresponds to a specific amount of another quantity. A unit rate is a special type of rate where the denominator is 1, allowing for easy comparison of quantities on a per-unit basis. For example, if a car travels 240 miles in 4 hours, the rate is \(\frac{240 \text{ miles}}{4 \text{ hours}}\), and the unit rate is \(\frac{60 \text{ miles}}{1 \text{ hour}}\) or 60 miles per hour (mph). Unit rates are particularly useful for comparing different quantities and making proportional calculations, as they standardize the comparison to a single unit of measurement.

核心公式

• \(\text{Rate} = \frac{\text{Quantity 1}}{\text{Quantity 2}}\)
• \(\text{Unit Rate} = \frac{\text{Quantity 1}}{1 \text{ unit of Quantity 2}}\)
• \(\text{Unit Rate} = \frac{\text{Rate}}{\text{Denominator}}\)
• \(\text{Price per unit} = \frac{\text{Total Price}}{\text{Number of Units}}\)
• \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

易错点

• ⚠️ Forgetting to include units in the final answer or mixing up which quantity should be in the numerator versus denominator, leading to inverted rates (e.g., writing hours per mile instead of miles per hour)
• ⚠️ Failing to simplify the rate to a unit rate correctly, such as not dividing both numerator and denominator by the same value or stopping the simplification process prematurely
• ⚠️ Confusing the concept of rate with ratio by not recognizing that rates must involve different units, or incorrectly applying unit rate calculations to problems that involve ratios of the same unit
• ⚠️ Making computational errors when converting between different units (e.g., forgetting conversion factors like 60 minutes in an hour or 12 inches in a foot) before calculating the unit rate